The motion of a spring 
I have a question about the force set by this spring, I saw many times that $\overrightarrow{F}=-Kx\overrightarrow{i}$.
I'm asking why not using $\overrightarrow{F}=Kx\overrightarrow{i}$ without the minus.
And supposing that $\overrightarrow{F}$ is changing during the motion of the solid object as well as the form of the spring.
Here $K$=the Hooke's constant, $K>0$.
P.S:I need a detailed explanation of this phenomenon.
 A: $x$ measures the difference in length of the spring in relation to its relaxed state. If you increase the length (positive $x$), the spring creates a force in the negative $x$ direction, because it wants to return to its relaxed state. Accordingly, if you compress the spring (negative $x$) the spring wants to expand (force in positive $x$ direction) in order to be relaxed again. That's why the negative sign in
$$ \vec F = - K x \vec i$$ gives the physically appropriate behaviour.
A: If the force exerted by the spring on the attached object / the acceleration of the object is in the same direction as its displacement, you can imagine that the object will continue to go to infinity because there is no opposite force bringing the object back to the equilibrium position. Hence, the minus sign give us the sense that the acceleration of the object always opposes its displacement.
A: The negative sign indicates that the force exerted by the spring is restorative.  This means that it is always in opposition to the direction of displacement.
Hooke's constant is a ratio which gives the force exerted per unit displacement from equilibrium.  It is unique to each spring.
